The demixing of a two-component fluid can be understood
as a gradient flow driven by interfacial energy and
limited by viscous dissipation. Bounds on the steepness
of the energy landscape translate into bounds on the
demixing rate. In order to understand the steepness of
the energy landscape one has to understand the distance
``in the large'' on configuration space given by the dissipation
metric ``in the small''.
It turns out that a transportation distance with logarithmic cost
is a good proxy for this distance. This observation builds on a quantitative
treatment of the DiPerna-Lions theory by DeLellis-Crippa.